Math 511
Operator theory and applications

Instructor Information Instructor: Richard Froese Email: rfroese-at-math-dot-ubc-dot-ca Office: Math Annex 1106 Hours: By appointment Phone: 604-822-3042

Course Information Section 101 Tuesday and Thursday 14:00-15:20 MATH 202 This page http://www.math.ubc.ca/~rfroese/math511 will be updated throughout the term.

Problem sets I will post problem sets here periodically throughout the term. Your grade in the course will be based on these. Problem set 1 due Tuesday September 24 Solutions 1 Problem set 2 due Tuesday October 22 Solutions 2 Problem set 3 due Thursday November 28 Solutions 3

Prerequisites A course in measure theory at the level of UBC's Math 420/507, and the basics of Hilbert and Banach spaces (which we will review).

(Optional) Text Reed and Simon, Methods of Modern Mathematical Physics, Vol I: This is an excellent book, but very expensive, so it is not required.

Notes Notes by Joel Feldman covering many topics in this course. Notes by Eugene Kritchevski on bounded Borel functions and the uniqueness of the Borel functional calculus.

Topics 1. Review of Hilbert spaces and Banach spaces Definitions, examples, strong and weak convergence. 2. Bounded operators on Hilbert space Topologies, adjoints, self-adjoint operators, resolvents and spectrum, spectral radius, unitary operators, (partial) isometries, positive operators, polar decomposition, spectral theorem for bounded self-adjoint operators. 3: Unbounded operators Closed operators, extensions, adjoints, resolvents and spectrum, symmetric and self adjoint operators, spectral theorem, unitary groups and Stone's theorm, quadratic forms 4: Compact operators Definitions, analytic Fredholm theorem, trace ideals, trace, determinant and Lidski's theorem, g(p)f(x)